transcendence degree

English

Noun

transcendence degree (plural transcendence degrees)

1. (algebra, field theory, of a field extension) Given a field extension L / K, the largest cardinality of an algebraically independent subset of L over K.
   • 2004, F. Hess, An Algorithm for Computing Isomorphisms of Algebraic Function Fields, Duncan Buell (editor), Algorithmic Number Theory: 6th International Symposium, ANTS-VI, LNCS 3076, page 263,

     Let $F_{1}/k$ and $F_{2}/k$ denote algebraic function fields of transcendence degree one.

   • 2007, Anthony W. Knapp, Advanced Algebra, Springer (Birkhäuser), page 422:

     Lemma 7.19 Suppose that $L$ is a field extension^{[sic – meaning extension field]} of transcendence degree $r$ over a field $K$ and that $L$ is not separably generated over $K$. If $x_{1},\dots ,x_{n}$ are elements of $L$ such that $L=K(x_{1},\dots ,x_{n})$, then for a suitable relabeling of the $x_{i}$'s, the subfield $K(x_{1},\dots ,x_{r+1})$ of $L$ is of transcendence degree $r$ and is not separably generated over $K$.

   • 2008, Bernd Sturmfels, Algorithms in Invariant Theory, 2nd edition, Springer, page 24:

     Proposition 2.1.1 Every finite matrix group $\Gamma \subset \operatorname {GL} (\mathbb {C} ^{n})$ has $n$ algebraically independent invariants, i.e., the ring $\mathbb {C} [x]^{\Gamma }$ has transcendence degree $n$ over $\mathbb {C}$.

Usage notes

• A transcendence degree is said to be of a field extension (i.e., $L/K$). More properly, it is the cardinality of a particular type of subset of the extension field $L$, although the context of the field extension is required to make sense of the definition.
• Relatedly, a transcendence basis of $L/K$ is a subset of $L$ that is algebraically independent over $K$ and such that $L$ is an algebraic extension of $K(S)$ (that is, $L/K(S)$ is an algebraic extension).
  • It can be shown that every field extension $L/K$ has a transcendence basis, whose cardinality, denoted $\operatorname {trdeg} _{K}L$ or $\operatorname {trdeg} (L/K)$, is exactly the transcendence degree of $L/K$.

Synonyms

• (cardinality of largest algebraically independent subset of a given extension field): transcendental degree

Related terms

• transcendence
• transcendence basis

Translations

cardinality of largest algebraically independent subset of a given extension field

See also

• algebraically independent

Further reading

• Algebraic independence on Wikipedia.Wikipedia
• Algebraic independence on Encyclopedia of Mathematics
• Transcendence Degree on Wolfram MathWorld